Then w will receive a small increment Q^ Q^ or 



dz 

 tiere dw/dz and 8z are complex numbers; they can be written 



dw 

 dz 



= Re'^, 8z = \8z\ e'' 



in terms of real numbers R, a, \Sz\ and (3. Then 



8w = /?|§3|c'(^ +°> 



Suppose that R ^ so that dw/dz /■ 0. Then this last equation shows that the line 

 element 8w can be formed out of the line element 8z by first stretching or shrinking it in the 

 ratio represented by R, or by the modulus of dw/dz, and then rotating it through the angle a , 

 which is the amplitude of dw/dz. 



Thus the vector representing 8w makes with the axis of reals on the w)-plane an angle 

 greater by a than the angle that the vector representing 8z makes with the real axis on the 

 2-plane; see Figure 22. Any other line element at P., such as P. P,, is changed in scale in 

 the same ratio and is rotated through the same angle and in the same direction. The derivative 

 dw/dz can be thought of as an operator that transforms the line elements in this manner; it 

 stretches the local area in the ratio R and rotates it through the angle a . 



It follows that, if the two curves interesect at an angle y on the 2-pIane, the transformed 

 curves will intersect at the same angle y on the w-plane. Furthermore, the angle is not turned 

 over; a rotation in the same direction through an angle y swings the tangent from one curve to 

 the other on either plane. Thus a transformation by means of a regular function /(s) completely 

 preserves the angles between intersecting curves at all points at which df/dz ^ 0. Infinitesimal 

 figures also keep the same shape, although they may be changed in scale and rotated through a 

 certain angle, without being turned over. 



Figure 22a Figure 22b 



Figure 22 — Illustration of conformal mapping. 



40 



